The present invention generally to optical communications, and more particularly, to a system and method for determining how much of the optical spectrum of a given transmission link can be used to provision wavelengths without the need for polarization mode dispersion (PMD) compensation.
Optical communications have revolutionized the telecommunications industry in recent years. The fiber optic medium provides the ability to efficiently transmit high bit rate signals through a low-loss medium. The development of modern high bandwidth techniques, and wavelength division multiplexing (WDM) to permit the simultaneous transmission of multiple high bandwidth channels on respective wavelengths, has enabled a tremendous increase in communications capacity. The last decade has been seen efforts to increase capacity by taking advantage of the fiber optic medium to the maximum extent possible.
Signals transmitted through an optical medium can be affected by PMD, which is a form of signal distortion that can be caused by subtle physical imperfections in the optical fiber. In principle, an optical fiber with a circular core has rotational symmetry, so that there is no preferred direction for the polarization of the light carrying the optical signal. However, during fabrication, jacketing, cabling, and installation, perturbations in the fiber that will distort this symmetry can occur, thereby causing the fiber to “look different” to various optical polarizations. One of the manifestations of this loss of symmetry is “birefringence,” or a difference in the index of refraction for light that depends on the light's polarization. Light signals with different polarization states will travel at different velocities. In particular, there will be two states of polarization (SOPs), referred to as the “eigenstates” of polarization corresponding to the asymmetric fiber. These eigenstates form a basis set in a vector space that spans the possible SOPs, and light in these eigenstates travels at different velocities.
A birefringent optical fiber transporting a modulated optical signal can temporally disperse the resulting optical frequencies of the signal. For example, an optical pulse, with a given optical polarization, can be formed to represent a “1” in a digital transmission system. If the signal is communicated through a medium with uniform birefringence (i.e., remaining constant along the length of the fiber), the SOPs can be de-composed into corresponding eigenstates, thereby forming two independent pulses, each traveling at its own particular velocity. The two pulses, each a replica of the original pulse, will thus arrive at different times at the end of the birefringent fiber. This can lead to distortions in the received signal at the end terminal of the system. In this simple illustrative case, the temporal displacement of the two replicas, traveling in the “fast” and “slow” SOPs, grows linearly with distance.
In a typical optical communications system, birefringence is not constant but varies randomly over the length of the transmission medium. Thus, the birefringence, and therefore, the eigenstate, changes with position as the light propagates through the length of the fiber. In addition to intrinsic changes in birefringence resulting from imperfections in the fabrication processes, environmental effects such as, for example, temperature, pressure, vibration, bending, etc., can also affect PMD. These effects can likewise vary along the length of the fiber and can cause additional changes to the birefringence. Thus, light that is in the “fast” SOP in one section of fiber might become be in the “slow” SOP at another section of the fiber. Instead of increasing linearly with distance, the temporal separations in the pulse replicas eventually take on the characteristics of a random walk, and grow with the square root of the distance. Despite the local variations in the fast and slow states, it is understood that when the fiber as a whole is considered, another set of states can be defined that characterize the PMD for the entire fiber and split the propagation of the signal into fast and slow components. These “principal states” can be imaged (in a mathematical sense) back to the input face, and used as an alternative basis set. Thus, an arbitrary launch SOP will have components in each of the principal states, and distortion will result from the replication of the pulses after resolution into principal states and their differential arrival times. While the physical process is described in the foregoing in a “global” as opposed to “local” sense, the basic impairment is the same; distortion results from the time delay introduced in the pulse replicas.
The above discussion relates to “narrowband” signals, i.e., having a narrow enough bandwidth that the optical properties of the fiber can be characterized as operating at a single wavelength. This is commonly referred to as “first order PMD.” Birefringence, however, can also vary with wavelength, such that each section of fiber may have slightly different characteristics, both in the magnitude and direction of the birefringence. As a consequence, after a long propagation through an optical medium, light from two neighboring wavelengths initially having the same polarization may experience what looks like a fiber with two different characteristics.
Theoretically, PMD can be represented by a Poincare sphere, or “Stokes' space” representation. In this representation, the equations of motion for SOPs and PMD at a given optical frequency are given by:∂s/∂z=β×s  (1a)∂s/∂ω=τ×s  (1b)∂τ/∂z=∂β/∂ω+β×τ  (1c)In these equations (which are in the “representation” space, not “real” space) “β” represents the birefringence of the fiber at position z, “s” represents the SOP of the light at position z, and “τ” represents the PMD. Generally, Eqn. (1a) states that birefringence causes the representation of the SOP to rotate about the “β” axis as light propagates through the fiber. Eqn. (1b) states that, when viewed at a given position (e.g., the fiber output), the system's PMD causes the SOP to rotate about it as a function of optical frequency. In this regard, light launched at a given optical frequency will evolve to an SOP at the output, and if the optical frequency is then changed (but the launch polarization remains the same), the SOP at the output will also begin to rotate about the PMD vector, τ. Eqn (1c) states that the vector characterizing PMD changes along the length of the fiber. The driving term in Eqn (1c), β′=∂β/∂ω, which we refer to as the “specific PMD,” describes the relationship of birefringence to optical frequency. Even for the simplest cases, there is usually a non-zero driving term (and thus PMD) for birefringent fibers. Based on the above, the vector s will suffer infinitesimal rotations about the axis defined by β, and that the rotation axis will change as β changes with distance (and parametrically with time). However, the total evolution of s can be represented by a single, finite rotation based upon Euler's theorem. If the signal bandwidth is large enough to experience these variations, it is commonly referred to as “higher order” PMD. Higher order PMD also leads to pulse distortion as the optical bandwidth of the signal increases. As the bandwidth increases, the input signal can be decomposed into Fourier components, with each propagated in accordance with the equations discussed above, and the components collected at the output. In the narrowband context, for illustrative purposes, the “concatenation rule” represented by the above equations states that the PMD of a given section of fiber can be “imaged” to the PMD at the output through the same transformation that governs birefringence. For a fiber consisting of two sections having respective PMDs τ1 and τ2, and respective rotations of the SOP via finite rotations R1 and R2, the total PMD can be represented by:τ=τ2+R2τ1  (2)This equation states that the final PMD vector is represented by the vectorial sum of the second (i.e. final) section's PMD vector and the first section's PMD vector, but only after that first PMD vector has been rotated by the same rotation operator (R2) that rotates the SOPs propagating at that wavelength. This is shown by noting the rotations by β implied in Eqns. 1a and 1c.
A generalization of Eqn. 2 shows that a similar rule applies for a fiber having multiple sections. Thus, each section of length Δz can be considered as having it's own uniform primitive PMD vector, β′Δz. The PMD of the entire multi-sectioned fiber can be characterized as a vector sum of the transformed primitive PMDs, one for each section, where each PMD primitive vector is transformed by the concatenated rotation of all the sections between it and the output. Since each of these constituent vectors is only a transformed version of its corresponding primitive PMD vector, each has the same length as its primitive vector, but effectively suffers a random rotation (the Euler's theorem equivalent of the concatenated rotations between the section and the output). This process is illustrated in FIG. 1, where for an arbitrary optical frequency ω0, the fiber (hereinafter, the optical fiber will be referred to as optical fiber) 100 is segregated into five independent sections (i.e., A, B, C, D, E), where each section's PMD is represented by a vector (row 102) directly below that section, and these PMD vectors represent a random distribution in magnitude and direction for the respective sections of the optical fiber. Each section's PMD vector (except the last one's) is imaged to the end and is shown on the right side of the figure (at 106) as a primed version of the original. Thus, the PMD vector for section B is propagated through sections C, D, and E, resulting in its output image, vector B′. The PMD for the entire fiber is then computed as the vector sum of these constituents as depicted at 108 in FIG. 1.
Referring now to FIG. 2, the PMD of the same fiber is shown at a slightly different optical frequency, ω0+Δω. In this example, in row 202 the PMD for each section at ω0 (from FIG. 1) is represented by dotted vectors, while the PMD for each section at ω0+Δω is represented by solid vectors. Each primitive vector corresponding to this neighboring frequency (ω0+Δω) is slightly different than the primitive vector for the original frequency ω0. This, by itself, results in a slightly different sum for the total PMD vector at ω0+Δω. However, in addition to slight changes in the primitive vectors, the new optical frequency also causes different rotations in each section, since the birefringence in each section can also be a function of optical frequency. The images for each section are imaged (trajectories 204) to the output at 206, and are slightly different from those depicted in FIG. 1 as shown by the difference at 206 between the solid and dotted arrows. These change more dramatically as the optical frequency changes. In FIG. 2, the total PMD vector 208 at this new optical frequency is shown as a solid arrow, while the PMD vector at ω0 (from FIG. 1) is depicted as a dotted arrow. Thus, the PMD will change in magnitude and direction as a function of the optical frequency, even though the constituent PMD vectors for the sections may be drawn from the same statistical ensemble representing the fiber's properties. In large part, the study of PMD is a study of the properties of the statistics of the vector sum of these images.
Both the magnitude of the PMD vector, called the “differential group delay” or DGD, and the directions of the unit vectors parallel and anti-parallel to the PMD vector, called the “Principal States of Polarization” (PSPs), change with optical frequency. The principal states are orthogonal and thus are on opposite sides of the sphere. The unit vector is usually associated with the slowest mode. Most frequently, it is the DGD which is plotted in discussions of PMD, but variations in the PSPs with optical frequency also can cause distortion in the optical link. The properties of the PMD are therefore going to follow the statistics of the sum of a set of vectors from the sections of the fiber that are chosen from a distribution and then, for the most part, randomly rotated after propagation through the fiber before being summed.
It has been shown through experiments and simulations that differential group delay (DGD) statistics across channels is well approximated by a Maxwell distribution, resulting from many degrees of freedom. It is often assumed that the same statistics hold for every individual frequency, or channel, in a fiber if observed for a sufficient time, over which different paths of a 3-d random walk are realized. While this may be true for extremely long timescales such timescales may not be relevant compared to the duration of system operation. Nonetheless, at present the mean PMD of fiber routes is often chosen so that the probability of exceeding some maximally allowed DGD value is less than 10−5 with an assumption of Maxwellian statistics.
Many recent experimental studies on PMD were devoted to its temporal dynamics. There seems to be general agreement that in relatively short (<100 km) buried routes, the PMD changes only insignificantly over weeks and even months, essentially remaining ‘frozen’ over these timescales. At the same time significant variations of PMD have been observed in longer amplified routes, consisting of multiple buried fiber spans. These PMD variations are attributed to polarization rotations in the amplifier huts due to indoor temperature variation. Dispersion compensating modules, for example, were found to produce a full rotation in Stokes space when heated by 1-2° C.
More recent experimental data analysis shows that individual channels in a long system have strikingly different DGD statistics, with the mean DGD and its standard deviation varying by a factor of two across wavelength. Moreover, we see evidence linking these differences to the number of rotation points, or ‘hinges’ (such as the number of amplifier huts and bridges) in the system. These findings have significant implications for the statistics of system outages.